Basic mathematics teaching and learning aid

ABSTRACT

A basic mathematics teaching and learning aid comprises of nine arrays of square spaces placed with powers of 10 from 10 −4  to 10 6 , numbers formed by repeated addition of some of those powers of ten, and other numbers. Dot and dash are in some of square spaces. Numbers, sign, arrows, mathematical symbols, percentages, equations and condition are in open spaces among the arrays. Formation, presentation and relations among these numbers, dots, dashes, sign, arrows, mathematical symbols, percentages, equations and condition are primarily used as an aid in teaching and learning fundamental areas of basic mathematics which mainly includes numbers in base-ten number system, addition, subtraction, multiplication, division, relations of a number of n with zero (number 0) in mathematical operations, whole numbers, fractions, and percentage.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is entitled to the benefit of Provisional PatentApplication No. 60/201,334 filed on May 02, 2000.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] Not Applicable

REFERENCE TO A MICROFICHE APPENDIX

[0003] Not Applicable

BACKGROUND OF THE INVENTION

[0004] The present invention relates to an educational aid used inteaching and learning basic mathematics.

[0005] Prior art known to Applicant discloses that there has been manyinventions of appliances, devices, apparatuses, aids . . . which havebeen used to help teach and learn basic mathematics. Following are someof most related inventions which may be of interest regarding thepresent invention

[0006] “Educational Appliance”, U.S. Pat. No 1,163,125 to A. Bechmann inDec. 7, 1915.

[0007] “Educational Device”, U.S. Pat. No 1,400,887 to H. Liebman inDec. 20, 1921.

[0008] “Method of Teaching Mathematics”, U.S. Pat. No 4,445,865 to J.Sellon in May 1, 1984.

[0009] “Mathematics Teaching Device”, U.S. Pat. No 5,759,041 to B.Batten in Jun. 2, 1998.

[0010] “Flexible Planar Apparatus”, U.S. Pat. No 5,997,305 to L. Manglesin Dec. 7, 1999.

[0011] All of above inventions present something similar to times tablewith indicators (sticks, pins, colored transparent strips, or tip of apencil . . . ) or hand folding (U.S. Pat No. 5,997,305 ) used to locateproduct for two given factors. Some of inventions above are also used toteach addition, subtraction and division.

[0012] While times table and similar devices have been known and usedfor a long time, and partially accomplished some of objects andadvantages in teaching and learning basic mathematics, they present anumber of drawbacks. First, times table and similar devices requireusers to know concepts of horizontal, vertical and intersection or/anduse indicators (sticks, pins, colored transparent strips . . . ) inorder to find product of two factors. This requirement obviously makesit harder for the users to learn basic mathematics, especially for theyoung ones, for they have to learn many new things at the same time.Moreover, horizontal, vertical and intersection are concepts in higherlevel of mathematics, so it is not suitable to use them in teachingbasic mathematics. Second, times table and similar devices show theusers how to memorize but do not promote how to basically understandnumbers, mathematical operations and relationship among them. Forexample, the users memorize that product of two factors 3 and 2 is 6just because number 6 locates at intersection of “3” row and “2” row.Third, times table and similar devices have been primarily used to teachmultiplication, maybe division also. Thus, they are proved not to beeffective in teaching and learning other areas of basic mathematics.Presented are just some of the drawbacks of times table and similardevices.

[0013] Accordingly, it is seen that there exists a need to have a neweducational aid which will overcome the drawbacks of the times table andsimilar devices, and be efficient in teaching and learning fundamentalareas of basic mathematics.

BRIEF SUMMARY OF THE INVENTION

[0014] The present invention is based on base-ten number system withpower of ten “10 ^(a)” where “a” is an integer with range from minusinfinitive to plus infinitive.

[0015] Powers of ten locate in a plane in such a way that they form anarray with powers of ten consecutively ranging from minus infinitive toplus infinitive in directions from right to left on horizontal rows, andfrom bottom to top on vertical rows. Preferred embodiment of the presentinvention is based on said plane and framed by 6×6 array of powers often 10⁶ 10⁵ 10⁴ 10³ 10² 10¹ 10⁵ 10⁴ 10³ 10² 10¹ 10⁰ 10⁴ 10³ 10² 10¹ 10⁰10⁻¹ 10³ 10² 10¹ 10⁰ 10⁻¹ 10⁻² 10² 10¹ 10⁰ 10⁻¹ 10⁻² 10⁻³ 10¹ 10⁰ 10⁻¹10⁻² 10⁻³ 10⁻⁴

[0016] Powers of ten in the above array are then expanded, regrouped andplaced in arrays of square spaces which are separated from one anotherwith same distance. Nine arrays with rows and columns of square spacesare arranged as below, and referred as bottom right, bottom, bottomleft, right, center, left, top right, top and top left arrays, in orderfrom right to left and from bottom to top, respectively: (2 × 2)  (2 ×10) (2 × 2) (10 × 2)  (10 × 10) (10 × 2)  (2 × 2)  (2 × 10) (2 × 2)

[0017] Formation, presentation and relations of numbers or numbers, dotsand dashes inside and among square spaces of the arrays as well as withnumbers, mathematical symbols, arrows, percentages, sign, equations andcondition in open spaces among said arrays are used in teaching andlearning fundamental areas of basic mathematics which mainly includesnumbers in base-ten number system (1, 2, 3, . . . ), addition,subtraction, multiplication, division, relations of a number with zero(number 0) in mathematical operations, whole numbers, fractions, andpercentage.

[0018] Following are advantages of the present invention regarding todisadvantages of the related inventions identified in section‘Background Of The Invention’:

[0019] In stead of using indicators (sticks, pins, . . . ), hand foldingor/and concepts of vertical, horizontal and intersection to locateproduct of two factors, as with times table or similar devices, lines ofsame small numbers in center array of the preferred embodiment placesmall numbers, used as factors, next on the right and on the bottom ofbig number, used as product, in each square space. Product of twofactors is not just accepted and memorized but it can be reasoned andproved. For example, big number 6 and 2 small numbers 2 and 3 on itsright and on its bottom in same square space can be used in 6=2×3 (2 and3 as factors ) or 6=2+2+2 (2 as addend with notice that 2 is added 3times to make 6) or 6=3+3 (3 as addend with notice that 3 is added 2times to make 6 ). Also, 6=4+2 and 6=8−2 are resulted from subtractionof big numbers 4 and 8 with small number 2's in right and left squarespaces of same row. Number 6 is also resulted from subtraction andaddition of numbers in top and bottom square spaces of same column.Times of number 2 in [6=2+2+2] and times of number 3 in [6=3+3] can alsobe used to present 6/3=2 and 6/2=3.

[0020] Thus, big number 6 with two small numbers 2 and 3 in same squarespace are not just used to be accepted and memorized only as “6 isproduct of 2 and 3”, but are partially used to promote a basicunderstanding of relationship among numbers 6, 2, 3, 4, 8 . . . ,especially 6, 2, 3, and among operations 3×2=6 , 2+2+2=6 , 3+3=6 , 4+2=6, 8−2=6 , 6/3=2 and 6/2=3. Advantages of the present invention regardingto disadvantages of identified related inventions are obvious, and inthese respects, the present invention substantially departs from theconventional concepts, designs, and methods of use of the prior art, toprovide a new educational aid used in teaching and learning fundamentalareas of basic mathematics.

[0021] Accordingly, besides objects and advantages described above,several objects and advantages of the present invention are:

[0022] (a) to provide a new educational aid;

[0023] (b) to provide a new educational aid which can be used inteaching and learning fundamental areas of basic mathematics;

[0024] (c) to provide a basic mathematics teaching and learning aidwhich can be used to promote minimum requiring of necessary means forteaching and basic mathematical knowledge for learning fundamental areasof basic mathematics;

[0025] (d) to provide a basic mathematics teaching and learning aidwhich can be used to promote basic understanding of relationship amongnumbers in base-ten number system as well as among basic mathematicaloperations;

[0026] (e) to provide a basic mathematics teaching and learning aidwhich can be used to promote basic understanding of meanings of basicmathematics operations;

[0027] (f) to provide a basic mathematics teaching and learning aidwhich can be used to promote basic preparation for teaching and learningmathematics at a higher level than basic mathematics.

[0028] (g) to provide a basic mathematics teaching and learning aidwhich can be manufactured with ease, readily available material, lowcost and convenience for packaging and transporting;

[0029] Other objects and advantages of the present inventions are toprovide a basic mathematics teaching and learning aid which is simple,compact, portable, reliable, inexpensive to the buying public, and canbe used to promote wide popularizing of basic mathematics masseducation.

[0030] Further objects and advantages of the present inventions willbecome apparent from a consideration of the accompanying drawings andfollowing description.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

[0031]FIG. 1 shows an isometric view of the preferred embodiment of thepresent invention.

[0032]FIG. 2 shows a plan view of the preferred embodiment of thepresent invention.

[0033]FIG. 2A shows a plan view of the preferred embodiment withreference numerals.

[0034]FIG. 2B shows portion of squares, each defined by four powers often, “10^(a)”, “10^(a+1)”, “10^(a+2)”, and “10^(a+1)” in a plane.

[0035]FIG. 2C shows symbol of “³10^(a)” presenting powers of ten inthree-dimensional space.

[0036]FIG. 3 shows a plan view of first alternative embodiment.

[0037]FIG. 4 shows a plan view of second alternative embodiment.

DETAILED DESCRIPTION OF THE INVENTION

[0038] (a) Reference Numerals In FIG. 2A:

[0039]12, 14, 16: bottom right, bottom, and bottom left arrays with(2row×2column), (2row×10column), and (2row×2column) of square spaces,respectively.

[0040]22, 24, 26: right, center, and left arrays with (10row×2column),(10row×10column), and (10row×2column) of square spaces, respectively.

[0041]32, 34, 36: top right, top, and top left arrays with(2row×2column), (2row×10column), and (2row×2column) of square spaces,respectively.

[0042]44: first columns of square spaces from right of arrays 14, 24,and 34.

[0043]20: first rows of square spaces from bottom of arrays 22, 24, and26.

[0044]64: intersection or common square space of column 44 and row 20.

[0045]40: open space among arrays, not in any array.

[0046]42: open space between column 44 and arrays 12, 22, and 32

[0047]18: open space between row 20 and arrays 12, 14, and 16

[0048] (b) Description of the Preferred Embodiment of the PresentInvention: (FIG. 1, FIG. 2, FIG. 2A, and FIG. 2B).

[0049] The preferred embodiment of the invention is illustrated in FIG.1(isometric view), FIG. 2 (plane view ), and FIG. 2A (plan view withreference numerals ). The preferred embodiment is made of flexibleplastic or suitable material with suitable length, height, and thicknessto carry information as illustrated in FIG. 1, FIG. 2, and FIG. 2A.

[0050] In portion on the right of array 24, each of ten square spaces ofcolumn 44 is placed at center with each of ten numbers from 1, 2, 3, . .. to 10 in direction from bottom to top, respectively. In each of ninesquare spaces of column 44, already having numbers from 2, 3, . . . to10, on top of each of those numbers, a pair of smaller numbers with adash (“-”) in between, from “2-1”, “3-1”, to “10-1” is placed,respectively . Dash is situated right on top of the big number, twosmall numbers are situated at top left and top right corners in eachsquare space. As for square space with big number 1, only a small number1, is placed at top right corner, in same square space. On bottom ofeach of big numbers from 1, 2, 3, . . . to 10, pair of a small numberand a period or dot (“.”), from “0 .”, “1 .”, “2 .”, . . . to “9 .”, isplaced with small number and dot situated at bottom left and bottomright corners in each square space, respectively.

[0051] In portion on bottom of array 24, except square space 64, each ofnine square spaces of row 20 is placed at center with each of ninenumbers from 2, 3, . . . to 10 in direction from right to left,respectively. In each of nine square spaces of row 20, already havingnumbers from 2, 3, . . . to 10, on top of each of those numbers, a pairof smaller numbers with a dash (“-”) in between, from “2-1”, “3-1”, . .. to “10- 1” is placed, respectively. Dash is situated right on top ofthe big number, two small numbers are situated at top left and top rightcorners in each square space. On bottom of each of big numbers from 2,3, . . . to 10, pair of a small number and a period or dot (“.”), from“1 .”, “2 .”, . . . to “9 .”, is placed with small number and dotsituated at bottom left and bottom right corners in each square space,respectively. Thus, only one small number or dash or dot is situated atone in 8 positions: on top, on bottom, on the right, on the left of thebig number, at bottom right, at top right, at top left, and at bottomleft corner in same square space.

[0052] Accordingly, except small numbers, dashes, and dots, big numbersin square spaces of column 44 and row 20, portions in array 24, exceptbig number 1, are formed by repeated addition by itself orarithmetically increasing of big number 1 (10⁰) in square space 64 upand to the left. If (big) number 10 is already in square space at theend of the column or row, then repeated addition by itself of (big)number 1 just fill the square spaces in between with (big) numbers from2 9, or continue to overwrite (big) number 10.

[0053] Big numbers in other square spaces of array 24 are also formed byrepeated addition by itself either to the left of each big number insquare spaces of column 44 in array 24 or up (or to the top) of each bignumber in square spaces of row 20 in array 24, except big number 1 insquare space 64. After the repeated addition, array 24 has four powersof ten 10⁰, 10¹, 10², 10¹, presenting in decimal form, 1, 10, 100, 10,in bottom right, top right, top left and bottom left square spaces,respectively.

[0054] In order to show how big numbers in square spaces of column 44 orrow 20 perform repeated addition to form other big numbers in squarespaces of array 24, lines of same small numbers in same row and columnare formed. These small numbers are placed on the right and on bottom ofbig numbers in square spaces of array 24.

[0055] Thus, each square space of array 24 has a big number at center, asmall number on the right and other small number on bottom of the bignumber. Same small numbers on the right of big numbers in same row aresame with big number in square space of column 44 in same row. Forexample: same small number 9's on the right of big numbers 90, 81, 72,63, 54, 45, 36, 27, 18 and 9, in same row, are same with big number 9 insquare space of column 44 in same row. Same small numbers on bottom ofbig numbers in same column are same with big number in square space ofrow 20 in same column. For example: same small number 3's on bottom ofbig numbers 30, 27, 24, 21, 18, 15, 12, 9, 6, and 3, in same column, aresame with big number 3 in square space of row 20 in same column.

[0056] Each big number in each square space of array 24, excluding bignumbers in square spaces of column 44 and row 20, is moved up and to theleft a little bit so that two other small numbers can comfortably fitwithin same square space.

[0057] Small numbers on the right and on bottom of the big number ineach square space of column 44 and row 20 in array 24 can be resized tofit within same square space.

[0058] In square space at top left corner of array 24, small number 10on the right of the big number 100 is moved downward a little bit to fitwithin same square space.

[0059] Four powers of ten 10², 10³, 10⁴ and 10³ are placed in foursquare spaces of array 26, in bottom right, top right, top left andbottom left square space, respectively. Bottom right and bottom leftsquare spaces are also in row 20. Powers of ten 10² and 10³ in bottomright and bottom left square spaces each performs repeated addition byitself up or to the top to form other numbers in other square spaces ofarray 26.

[0060] Four powers of ten 10², 10³, 10⁴ and 10³ are placed in foursquare spaces of array 34, in bottom right, top right, top left andbottom left square spaces, respectively. Bottom right and top rightsquare spaces are also in column 44. Powers of ten 10² and 10³ in bottomright and top right square spaces each performs repeated addition byitself to the left to form other numbers in other square spaces of array34.

[0061] Four powers of ten 10⁻², 10⁻¹, 10⁰ and 10⁻¹ are placed in foursquare spaces of array 22, in bottom right, top right, top left andbottom left square spaces, respectively. Bottom right and bottom leftsquare spaces are also in row 20. Powers of ten 10⁻² and 10⁻¹ in bottomright and bottom left square spaces each performs repeated addition byitself up or to the top to form other numbers in other square spaces ofarray 22.

[0062] Four powers of ten 10^(−2, 10) ^(−1, 10) ⁰, and 10⁻¹ are placedin four square spaces of array 14, in bottom right, top right, top leftand bottom left square spaces, respectively. Bottom right and top rightsquare spaces are also in column 44. Powers of ten 10⁻² and 10⁻¹ inbottom right and top right square spaces each performs repeated additionby itself to the left to form other numbers in other square spaces ofarray 14.

[0063] Four powers of ten 10⁻⁴, 10⁻³, 10⁻², and 10⁻³ are placed in foursquare spaces of array 12, in bottom right, top right, top left andbottom left square spaces, respectively.

[0064] Four powers of ten 10⁰, 10¹, 10², and 10¹ are placed in foursquare spaces of array 16, in bottom right, top right, top left andbottom left square spaces, respectively.

[0065] Array 32 is identical to array 16.

[0066] Four powers of ten 10⁴, 10⁵, 10⁶, and 10⁵ are placed in foursquare spaces of array 36, in bottom right, top right, top left andbottom left square spaces, respectively.

[0067] Excluding numbers in square spaces of column 44 and row 20,number in each square space of arrays 12, 14, 16, 22, 26, 32, 34, and 36is presented in two forms: decimal form and exponential form. Thisresults in having two numbers in each square space: upper number indecimal form and lower number in exponential form.

[0068] Same as other numbers in other square spaces of row 20 in array24, on top of each of two big numbers 10³ and 10² in two square spacesof row 20 in array 26, a pair of smaller numbers with a dash (“-”) inbetween, “10³-1” and “10²-1”, is placed, respectively. Dash is situatedright on top of the big number, two small numbers are situated at topleft and top right corners in each square space. On bottom of each ofbig numbers, pair of a small number and a period or dot (“.”), “999 .”and “99 .”, is placed with small number and dot situated at bottom leftand bottom right corners in each square space, respectively.

[0069] Two square spaces of column 44 in array 34 from bottom to top areidentical with two square spaces of row 20 in array 26 from right toleft, respectively.

[0070] On top of each of two big numbers 10⁻¹ and 10⁻² in two squarespaces of row 20 in array 22, a pair of smaller numbers with a dash(“-”) in between, “1-10¹” and “1-10⁻²”, is placed, respectively. Dash issituated right on top of the big number, two small numbers are situatedat top left and top right corners in each square space. At bottom leftcorner of each square space, a period or dot (“.”) is placed.

[0071] Two square spaces of column 44 in array 14 from bottom to top areidentical with two square spaces of row 20 in array 22 from right toleft, respectively.

[0072] Symbol of addition and arrow pointing to the left are placed inopen space 42, between arrays 22 and 24, next to the right of two squarespaces of column 44 with big numbers 5 and 6 at center. Another symbolof addition and arrow pointing to the top are placed in open space 18,between arrays 14 and 24, on bottom of two square spaces of row 20 withbig numbers 5 and 6 at center. Symbol of subtraction and arrow pointingto the right are placed in open space 40, between arrays 26 and 24, nextto the left of two square spaces of array 24 with big numbers 50 and 60at center. Another symbol of subtraction and arrow pointing to thebottom are placed in open space 40, between arrays 34 and 24, next ontop of two square spaces of array 24 with big numbers 50 and 60 atcenter. Symbol of multiplication is placed at center of intersection ofopen spaces 42 and 18, among arrays 12,14, 24, and 22. A sign, formed bytwo short vertical and horizontal lines situating at top left ofdivision symbol, and the division symbol are placed in open space 40,among arrays 24, 26, 36 and 34.

[0073] Percentages of 100 and 1 with a little horizontal line in betweenare placed in open space 40, right on bottom of square space with bignumber 10² of column 44. Other percentages of 100 and 1 with a littlevertical line in between, are placed in open space 40, next to the rightof square space with big number 10² of row 20.

[0074] Quotients resulted from “1” as dividend with each big number ineach square space of column 44 and row 20 as divisor, excluding squarespace 64, are placed in open space 42, next to each square space ofcolumn 44 on the right, and in open space 18, right on bottom of eachsquare space of row 20, accordingly. Each of numbers presenting thosequotients, except round ones, has four digits to the right of decimalpoint and a dash (“-”) replacing repeating digits, and is not roundedoff. Quotient resulted from “1” as dividend with big number 1 in squarespace 64 as divisor, which is 1, is placed at top left corner in squarespace 64 (small number 1).

[0075] Two equations, n+0=n and n−0=n, are placed on bottom of squarespace 64, in open space 18, the former on top and the latter on bottom.

[0076] Two equations, n×0=0 , 0/n=0, and condition n#0 for the latterequation are placed on the right of square space 64, in open space 42,in order, top, middle and bottom, respectively.

[0077] It is preferable:

[0078] On area of other suitable sides of the preferred embodiment,other information as in Table 2D can be placed. In ‘Table 2D’, quotientsresulted from “1” as dividend with each number from 1, 2, 3, . . . to10, as divisor are placed in first column on the right of “equal”symbols (“-”), on bottom and in same column with number 1. Multiples ofeach quotient are placed in same row, and each is placed on bottom andin same column with each number of 2, 3, . . . to 10, accordingly, ortill equal to one (1). Numbers presenting the quotients and theirmultiples, except round ones, have four digits to the right of decimalpoint and a dash (“-”) replacing repeating digits, and are not roundedoff. TABLE 2D Number zero (0) on the left of decimal point of each ofthe quotients and their multiples is omitted. 1 2 3 4 5 6 7 8 9 10 1 ÷10 = .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1 ÷ 9 = .1111− .2222− .3333− .4444−.5555− .6666− .7777− .8888− 1 1 ÷ 8 = .125 .25 .375 .5 .625 .75 .875 1 1÷ 7 = .1428− .2857− .4285− .5714− .7142− .8571− 1 1 ÷ 6 = .1666− .3333−.5 .6666− .8333− 1 1 ÷ 5 = .2 .4 .6 .8 1 1 ÷ 4 = .25 .5 .75 1 1 ÷ 3 =.3333− .6666− 1 1 ÷ 2 = .5 1 1 ÷ 1 = 1

[0079] Numbers, dashes, dots, mathematical symbols, arrows, percentages,sign, equations and condition in square spaces and in open spaces of thepreferred embodiment can be applied with different colors and sizes aswell as adjusted in such a way that facilitates understanding of theusers and is still within scope of the present invention.

[0080] (c) Description of First Alternative Embodiment

[0081]FIG. 3 and FIG. 2A)

[0082] From the preferred embodiment, one column of square spaces withbig numbers identical to big numbers of column 44 is placed next tocolumn 44 to the right One row of square spaces with big numbersidentical to big numbers of row 20 is placed next to row 20 on bottom.Thus, arrays 12, 22, 32, and everything in open spaces 42 have to moveto the right, and arrays 12, 14, 16, and everything in open spaces 18have to move downward so that open spaces 18 and 42 are still as before.Then, all small numbers at corners, dash, and dot in each square spaceof column 44 are moved to next square space to the right in same row insame positions. All small numbers at corners, dash, and dot in eachsquare space of row 20 are moved down to next square space on bottom insame column in same positions. Small numbers at corners and dot insquare space 64 with big number 1 are copied and moved two times, to theright and down. Each big number in each square space of row 20 andcolumn 44, portions in arrays 22, 26, 14, and 34 is then presented intwo forms: decimal form and exponential form. Thus, each square space ofrow 20 and column 44, portions in arrays 22, 26, 14, and 34 has twonumbers: upper number in decimal form and lower number in exponentialform. Number zero (0) is placed at center in new square space formed atintersection of new column and row with big numbers identical to bignumbers of column 44 and row 20.

[0083] Nine arrays with rows and columns of square spaces of the firstalternative embodiment with reference numerals are:

[0084]12, 14, 16: bottom right, bottom, and bottom left arrays with(2row×2column), (2row×11column), and (2row×2column) of square spaces,respectively.

[0085]22, 24, 26: right, center, and left arrays with (11row×2column),(11row×11column), and (11row×2column) of square spaces, respectively.

[0086]32, 34, 36: top right, top, and top left arrays with(2row×2column), (2row×11column), and (2row×2column) of square spaces,respectively.

[0087]44: first columns of square spaces from right of arrays 14, 24,and 34.

[0088]20: first rows of square spaces from bottom of arrays 22, 24, and26.

[0089]64: intersection or common square space of column 44 and row 20.

[0090]40: open space among arrays, not in any array.

[0091]42: open space between column 44 and arrays 12, 22, and 32

[0092]18: open space between row 20 and arrays 12, 14, and 16

[0093] Numbers, mathematical symbols, arrows, sign, percentage,equations and condition are adjusted. For example: Small number 1 at topleft corner of square space with big number 1 of column 44 in array 24is moved outside to open space 42, next to the right of same squarespace. Small number 1 at top left corner of square space with big number1 of row 20 in array 24 is moved outside to open space 18, next onbottom of same square space. Two equations, n+0=n and n−0=n, are movedto next on bottom of square space 64. Two equations, n×0=0 , 0/n=0, andcondition n#0 are moved down next on the right of square space 64.Percentages of 100 and 1 with a little horizontal line in between aremoved to next on bottom of square space with big number 10² of column 44in array 34. Other percentages of 100 and 1 with a little vertical linein between are moved down next on the right of square space with bignumber 10² of row 20 in array 26. Symbols of addition and subtraction,and arrows in open spaces are moved so that they are in same row andcolumn of square spaces with same small number 5's of array 24. Symbolof multiplication still at intersection of open spaces 18 and 42, amongarrays 12, 14, 24, and 22.

[0094] (d) Description of Second Alternative Embodiment

[0095] (FIG. 3, FIG. 4, and FIG. 2A)

[0096] Referring to first alternative embodiment (FIG. 3 and FIG. 2A )One column of square spaces with big numbers zero's (0's) is placed nextto column 44 to the left. One row of square spaces with big numberszero's (0) is placed next to row 20 on top. Thus, arrays 12, 22, 32 andeverything in open spaces 42 have to move to the right, and arrays 12,14, 16, and everything in open spaces 18 have to move downward so thatopen spaces 18 and 42 are still as before.

[0097] Nine arrays with rows and columns of square spaces of the secondalternative embodiment with reference numerals are:

[0098]12, 14, 16: bottom right, bottom, and bottom left arrays with(2row×2column), (2row×12column), and (2row×2column) of square spaces,respectively.

[0099]22, 24, 26: right, center, and left arrays with (12row×2column),(12row×12column), and (12row×2column) of square spaces, respectively.

[0100]32, 34, 36: top right, top, and top left arrays with(2row×2column), (2row×12column), and (2row×2column) of square spaces,respectively.

[0101]44: first columns of square spaces from right of arrays 14, 24,and 34.

[0102]20: first rows of square spaces from bottom of arrays 22, 24, and26.

[0103]64: intersection or common square space of column 44 and row 20.

[0104]40: open space among arrays, not in any array.

[0105]42: open space between column 44 and arrays 12, 22, and 32

[0106]18: open space between row 20 and arrays 12, 14, and 16

[0107] Square space 64 is now placed with symbol of multiplication.Equations, n+0=n and n−0=n, are in open space 18 and in same column withzero column. Equations, 0×n=0, 0/n=0 and condition n#0 are in open space42 and in same row with zero row.

[0108] In center of the plane with squares and each defined by fournumbers, “10^(a)”, “10^(a+1”), “10^(a+2)”and “10^(a+1)” at four corners,bottom right, top right, top left, and bottom left corner, respectively,two pairs of identical numbers of “10^(a)” with numbers of 0 in betweenintersect at location with number 0 at center, four numbers 0's at foursides, and four numbers of 10⁰ each presented as 1 at four corners withthe number 1 on bottom right corner replaced with symbol ofmultiplication, as shown below and in FIG. 4. 1 0 1 0 0 0 1 0 X

[0109] Powers of ten in a plane will be shown in detail at the end ofsection ‘Operation’ of the following.

[0110] (e) Operation:

[0111] **** Operation of The Preferred Embodiment (FIG. 2, FIG. 2A, andFIG. 2B).

[0112] Referring to FIG. 2 and FIG. 2A of the preferred embodiment ofthe present invention:

[0113] In array 24, big numbers from 1 to 10 in square spaces of column44 and row 20 are used as whole numbers. The big numbers 1 and 10 arealso considered as 10⁰ and 10¹, base 10 with powers 0 and 1. Big numbersand small numbers in square spaces of array 24 are used as whole numbersin:

[0114] (1) Addition operation for numbers in square spaces of same row:

[0115] The big number and small number on the right in each square spaceof array 24 are used as addends to make sum which is big number in nextsquare space to the left in same row, as defined by addition symbol andarrow pointing to the left in open space 42, between arrays 24 and 22.The small number on bottom of the big number in each square space isused to indicate times the small number on the right added together tomake sum which is the big number in same square space.

[0116] For example, referring to square space in sixth column from leftand fifth row from bottom, in the middle of array 24, with big number 25at center and two small number 5's on the right and on bottom, the smallnumber 5 on bottom is used to in indicate that five small number 5's onthe right of the big number are added together to make sum which is thebig number 25. The big number 25 is result from adding between bignumber 20 and small number 5 on the right of the big number 20, in nextsquare space on the right, in same row. After the adding, small number4, on bottom of the big number 20, is raised one and becomes smallnumber 5 on bottom of the big number 25, in next square space to theleft. Likewise, the big number 25 and small number 5 on the right areadded together to make sum which is big number 30 in next square spaceto the left, in same row. After adding, smaller number 5 on bottom ofthe big number 25 is raised one and becomes small number 6 on bottom ofthe big number 30, in next square space to the left. Four small number5's added together to make sum 20, five small number 5's to make sum 25,and six small number 5's to make sum 30, are shown in three squarespaces, from right to left in same row, respectively. Sum of the bignumber and small number on the right as addends in each square space offirst column from left of array 24 is not shown.

[0117] (2) Addition operation for numbers in square spaces of samecolumn:

[0118] The big number and small number on bottom in each square space ofarray 24 are used as addends to make sum which is big number in nextsquare space up on top in same column, as defined by addition symbol andarrow pointing to the top in open space 18, between arrays 24 and 14.The small number on the right of the big number in each square space isused to indicate times the small number on bottom added together to makesum which is the big number in same square space.

[0119] For example, referring to square space in first column from leftand fifth row from bottom of array 24 with big number 50 at center,small number 5 on the right and small number 10 on bottom, the smallnumber 5 on the right is used to in indicate that five small number 10'son bottom of the big number are added together to make sum which is thebig number 50. The big number 50 is result from adding between bignumber 40 and small number 10 on bottom of the big number 40 in bottomsquare space, in same column After the adding, small number 4, on theright of the big number 40, is raised one and becomes small number 5 onthe right of big number 50, in next square space up on top. Likewise,the big number 50 and small number 10 on bottom are added together tomake sum which is big number 60 in next square space up on top, in samecolumn. After adding, smaller number 5 on the right of the big number 50is raised one and becomes small number 6 on the right of the big number60, in next square space up on top. Four small number 10's addedtogether to make sum 40, five small number 10's to make sum 50, and sixsmall number 10's to make sum 60 are shown in three square spaces frombottom to top in same column, respectively. Sum of the big number andsmall number on bottom as addends in each square space of first row fromtop of array 24 is not shown.

[0120] (3) Subtraction operation for numbers in square spaces of samerow:

[0121] The big number and small number on the right in each square spaceof array 24 are used as minuend and subtrahend to make difference whichis big number in next square space to the right in same row, as definedby subtraction symbol and arrow pointing to the right in open space 40,between arrays 26 and 24 (The small number on bottom of the big numberin each square space is used to indicate times the small number on theright added together to make sum which is the big number in same squarespace).

[0122] For example, referring to square space in sixth column from leftand fifth row from bottom, near the middle of array 24 with big number25 at center and two small number 5's on the right and on bottom, thebig number 25 is difference resulted from subtracting between big number30 as minuend, and small number 5 on the right of the big number 30 assubtrahend, in next square space on the left in same row. After thesubtracting, small number 6 on bottom of the big number 30 is decreasedone and becomes small number 5 on bottom of the big number 25, in nextsquare space to the right. Likewise, the small number 5 on the right ofbig number 25 is subtracted from big number 25 to make difference whichis big number 20 in next square space to the right, in same row. Aftersubtracting, small number 5 on bottom of the big number 25 is decreasedone and becomes small number 4 on bottom of the big number 20 in nextsquare space to the right. 30 subtracts 5 to make 25. 25 subtracts 5 tomake 20. Three big numbers, 30, 25 and 20 are shown in three squarespaces from left to right, respectively. Difference of the big number asminuend and small number on the right as subtrahend in each square spaceof column 44 in array 24 is zero and not shown.

[0123] (4) Subtraction operation for numbers in square spaces of samecolumn:

[0124] The big number and small number on bottom in each square space ofarray 24 are used as minuend and subtrahend to make difference which isbig number in next square space on bottom, in same column, as defined bysubtraction symbol and arrow pointing to the bottom in open space 40,between arrays 34 and 24 (The small number on the right of the bignumber in each square space is used to indicate times the small numberon bottom added together to make sum which is the big number in samesquare space).

[0125] For example, referring to square space in first column from leftand fifth row from bottom of array 24 with big number 50 at center,small number 5 on the right and small number 10 on bottom, the bignumber 50 is result from subtracting between big number 60 as minuend,and small number 10 on bottom of the big number 60 as subtrahend, in topsquare space in same column. After subtracting, small number 6, on theright of the big number 60, is decreased one and becomes small number 5on the right of big number 50 in next square space on bottom in samecolumn. Likewise, the small number 10 on bottom of big number 50 issubtracted from big number 50 to make difference which is big number 40in next square space on bottom, in same column. After subtracting,smaller number 5 on the right of the big number 50 is decreased one andbecomes small number 4 on the right of the big number 40, in next squarespace on bottom in same column. 60 subtracts 10 to make 50. 50 subtracts10 to make 40. Three big numbers, 60, 50, and 40 are shown in threesquare spaces, from top to bottom in same column, respectively.Difference of the big number as minuend and small number on bottom assubtrahend in each square space of row 20 in array 24 is zero and notshown.

[0126] Addition and subtraction of numbers in the same row or column areused as repeated addition and repeated subtraction.

[0127] Two equations, n+0=n and n−0=n, placed in open space 18 on bottomof square space 64, are used to indicate that, for any number of n, nplus zero or n minus zero is still equal to number n, respectively.

[0128] (5) Multiplication operation

[0129] In each square space of array 24, small number on the right andsmall number on bottom of big number are used as factors, and the bignumber is used as product of these two factors. When compared withrepeated addition of numbers in same row, the small numbers on the rightand on bottom of the big number are multiplicand and multiplier,respectively. When compared with repeated addition of numbers in samecolumn, the small numbers on the right and on bottom of the big numberare multiplier and multiplicand, respectively. Then the big number isproduct of these multiplicand and multiplier.

[0130] The equation, 0×n=0, placed next to square space 64 in open space42, is used to indicate that, in multiplication operation, any number ofn multiplied with number zero (0), then the product of that number n andnumber zero is equal to zero.

[0131] Symbol of multiplication is placed at intersection of open spaces42 and 18, among arrays 24, 22,12, and 14.

[0132] (6) Division operation:

[0133] As an inverse process of multiplication, in each square space ofarray 24, big number is used as dividend and small numbers are used asdivisor and quotient. When compared with repeated subtraction of numbersin same row, the small numbers on the right and on bottom of the bignumber are divisor and quotient with meanings that each time smallnumber on the right is subtracted, small number on bottom is decreasedone to become small number on bottom of the big number in next squarespace to the right, in same row. In the same meanings, when comparedwith repeated subtraction of numbers in same column, the small numberson the right and on bottom of the big number are quotient and divisor.

[0134] The equation, 0/n=0, and condition, n#0, placed next to squarespace 64 in open space 42, are used to indicate that, in divisionoperation, if the divisor is not equal to zero, then the quotient isequal to zero when the dividend is equal to zero.

[0135] Symbol of division is placed in open space 40, among arrays 24,26, 36, and 34. The sign formed by two short vertical and horizontallines at top left of the division symbol is used to remind of two smallnumbers, on the right and on bottom of the big number, in each squarespace of array 24.

[0136] *** Numbers in square spaces of arrays 12, 14, 16, 22, 26, 32,34, and 36 can also be used in addition, subtraction, multiplication anddivision operations in an analogy way with big numbers and small numbersin square spaces of array 24.

[0137] *1 Addition and subtraction operations for numbers in squarespaces in same row or column Number in each square spaces of arrays 14,22, 26, and 34, except numbers in square spaces of column 44 and row 20,is presented in two forms in same square space, upper number in decimalform and lower number in exponential form. Lower numbers in squarespaces of first columns from left of arrays 14 and 34, and first rowsfrom top of arrays 22 and 26 are presented in the form of “10^(a+1)”.Big numbers in square spaces of outer right columns of arrays 14 and 34,and first rows from bottom of arrays 22 and 26, also in column 44 androw 20, are presented in the form of “10^(a)” with no upper decimalform.

[0138] For example: Numbers in top row of array 34 are presented as:

[0139] 10,000 9,000 8,000 . . . . . . 3,000 2,000

[0140] 10⁴ 9×10³ 8×10³. . . . . . 3×10³ 2×10³ 10³

[0141] If numbers (or number) in each square spaces of arrays 14, 22,26, and 34, excluding small numbers at corners (and dash, dot) in squarespaces of column 44 and row 20, are (or is) brought to the same form ofa number b multiplied by 10^(a) for lower number and a decimal form of(b×10^(a)) for upper number, then three numbers, “b” (number from 1, 2,3 . . . to 10), 10^(a) (a : integer {−∞, +∞}), and decimal form of(b×10^(a)), can be used in an analogy way with small numbers and bignumber in each square space of array 24 in addition and subtraction ofnumbers in same row or column.

[0142] For example: Numbers in top row of array 34 used in example abovecan be presented as:

[0143] 10,000 9,000 8,000 . . . . . . 3,000 2,000 1,000

[0144] 10×10³ 9×10³ 8×10³. . . . . . 3×10³ 2×10³ 1×10³

[0145] It is noticeable that small numbers on the right and on bottom ofbig number in each square space in array 24 are whole numbers while10^(a) in (b×10^(a)) is presented in exponential form.

[0146] Thus, three numbers, “b”, 10^(a), and decimal form of (b×10 ^(a))can also be used in multiplication and division operations of numbers insame square space.

[0147] *2 Multiplication and division operations:

[0148] In a similar way, if numbers in each square space of arrays 12,16, 32, and 36 are brought to the form (b×10^(a)) for lower number and adecimal form of (b×10^(a)) for upper number, then three numbers, “b”,10^(a), and decimal form of (b×10^(a)) can be used in an analogy way assmall numbers and big number in each square space of array 24 inmultiplication and division of numbers in same square space.

[0149] (7) Whole numbers:

[0150] Each big number in each square space of column 44 and row 20,portions in array 24, has a small number at bottom left corner in samesquare space. This number is used to remind of whole number(s) existing(if there is one ) between two big numbers in two square spaces next toeach other in same row or same column of array 24. This number is alsoused to remind of whole number(s) existing (if there is one ) betweennumber zero (0) (not shown ) and big number in each square space ofcolumn 44 and row 20 in array 24.

[0151] In first row from bottom of array 24, or row 20 in array 24, nowhole number exists between two any big numbers in two square spacesnext to each other in same row, and no whole number exists between zero(0) (not shown ) and big number 1, so at bottom left corner of squarespace with big number 1 of column 44 or square space 64, small numberzero (0) is placed. In second row from bottom of array 24, one wholenumber exists between two big numbers in two square spaces next to eachother. For example, on the right side of the row, one whole number,number 3, exists between two big numbers 2 and 4 in two square spacesnext to each other in same row; on the left side of the row, one wholenumber, number 19, exists between two big numbers 18 and 20 in twosquare spaces next to each other in same row; or one whole number,number 1, exists between number zero (0) (not shown ) and big number 2in square space of column 44 in same row. So, at bottom left corner ofsquare space with big number 2 of column 44 in same row, small number 1is placed. In top row of array 24, nine whole numbers exist between twobig numbers in two square spaces next to each other. For example, ninewhole numbers from 11, 12, 13, 14 . . . to 19 exist between two bignumbers 10 and 20, or nine whole numbers from 1, 2, 3, 4 . . . to 9exist between number zero (0) (not shown) and big number 10. So, atbottom left corner of square space with big number 10 of column 44 insame row, small number 9 is placed.

[0152] In the same manner, in first column from right of array 24, orcolumn 44 in array 24, no whole number exists between two any bignumbers in two square spaces next to each other in same column, and nowhole number exists between number zero (0) (not shown ) and big number1, so at bottom left corner of square space with big number 1 of row 20or square space 64, small number “0” is placed. In second column fromleft of array 24, eight whole number exists between two big numbers intwo square spaces next to each other in same column. For example, eightwhole numbers, from 82, 83, 84, . . . to 89, exist between big numbers81 and 90 on top of the column; or eight whole numbers, from 1, 2, 3, .. . to 8, exist between number zero (0) (not shown) and number 9, onbottom of the column. So, at bottom left corner of square space with bignumber 9 of row 20 in same column, small number 8 is placed.

[0153] ** Small numbers 99 and 999 at bottom left corners in lower andupper square spaces of column 44, portion on the right of array 34, areused in same way: 99 whole numbers exist between number zero (0) (notshown ) and number 100 or 10², and 999 whole numbers exist betweennumber zero (0) (not shown ) and number 1,000 or 10³. Between twonumbers in two square spaces next to each other in same row, 99 (inlower row), or 999 (in upper row ) whole numbers exist. Numbers insquare spaces of array 34 are whole numbers presented in decimal andexponential forms.

[0154] ** Small numbers 99 and 999 at bottom left corners in right andleft square spaces of row 20, portion on bottom of array 26, are used insame way as same numbers in square spaces of column 44 described abovefor numbers in square spaces of same column. Numbers in square spaces ofarray 26 are whole numbers presented in decimal and exponential forms.

[0155] (8) Fractions:

[0156] Each big number, except big number 1, in each square space ofcolumn 44 and row 20, portions in array 24, has a pair of small numberswith a dash in between on top of it in same square space. Numberspresented by that pair of small numbers are used with the big number toform fractions, proper fraction(s) and fraction with value equal to one.Denominators in those fractions are same as the big number andnumerators are numbers presented by that pair of small numbers. Forexample, numbers 1, 2, 3, 4 and 5 presented by pair of small numbers“5-1” on top big number 5, are used with the big number 5 to form properfractions ⅕, ⅖, ⅗, ⅘ and fraction {fraction (5/5)}. For square space 64with big numbers 1, the big number 1 and small number 1 at top rightcorner are used to form fraction {fraction (1/1)}, equal to 1 and placedat top left corner (small number 1) in same square space. Quotients indecimal forms of “1” as dividend with each big number in each squarespace of column 44, except square space 64, as divisor are placed nextto column 44, in open space 42, between arrays 24 and 22, accordingly;quotients in decimal forms of “1” as dividend with each big number ineach square space of row 20, except square space 64, as divisor areplaced next to row 20 on bottom in open space 18, between arrays 24 and14, accordingly. Numbers presenting these quotients and their multiples(in Table 2D), except round ones, have four digits to the right ofdecimal point and a dash (“-”), replacing repeating digits, and are notrounded off. These numbers can be used together with other numbers toget decimal forms of fractions and mixed numbers without using handcalculator in simple calculations. How these numbers are rounded offdepends on each need of users.

[0157] The fractions formed from big number and pair of small numbers ineach square space of column 44 and row 20 in array 24, except squarespace 64, are also used to combine with whole numbers of array 24 (shownand not shown) to form improper fractions and mixed numbers.

[0158] For example: 35×¼=35/4 (number 35 is in square space of fifth rowfrom bottom, seventh column from right of array 24)

[0159] 33+¾=33¾ (number 33 is not shown in array 24)

[0160] ** Pair of small numbers on top of big number in each squarespace of column 44 and row 20, portions in arrays 34 and 26respectively, is also used in the same way for fractions, except that,one small number of the pair and the big number in each square space arepresented in exponential form. Referring to other square spaces in samerow (for square spaces of column 44 in array 34), or in same column (forsquare spaces of row 20 in array 26), other similar numbers presented indecimal form as whole numbers can be found.

[0161] ** Pair of small numbers on top of big number in each squarespace of column 44 and row 20, portions in arrays 14 and 22respectively, is also used in the same way for fractions, except that,one small number of the pair and the big number in each square space arepresented in exponential forms which result in complex fractions withdecimal or fraction in the denominator or in both numerator anddenominator. For example, {fraction (1/10)}⁻¹, {fraction (1/0.1)},{fraction (0.1/0.1)}. Referring to other square spaces in same row (forsquare spaces of column 44 in array 14), or in same column (for squarespaces of row 20 in array 22), other similar numbers presented indecimal form can be found; or each quotient of {fraction (1/10)}⁻¹ and{fraction (1/10)}⁻² presenting complex fraction as whole number isplaced next to each square space on the right, in open space 42 (forsquare spaces of column 44 in array 14), and next to each square spaceon bottom, in open space 18 (for square spaces of row 20 in array 22),accordingly.

[0162] (9) Decimal point and percentage:

[0163] ** Decimal point:

[0164] Each big number in each square space of column 44 and row 20,excluding portions in arrays 14 and 22, has a dot (“.”) at bottom rightcorner in same square space. This dot is used to remind of decimal pointplaced on the right of ones digit and on the left of fraction part in adecimal number. In case no fraction part, decimal point is omitted. Thisperiod is also used as a reference for place value of digits on the leftand digits on the right as of decimal point. In square spaces with bignumbers 10⁻¹ and 10⁻² of column 44 in array 14 and of row 20 in array22, dots are placed at bottom left corners to indicate those big numberspresenting decimal fractions.

[0165] ** Percentage:

[0166] Percentages of 100 and 1, with a little horizontal line inbetween are placed in open space 40, near intersection of open spaces 40and 42, right on bottom of square space with big number 10² of column44. Other percentages of 100 and 1 with a little vertical line inbetween, are placed in open space 40, next to the right of square spacewith big number 10² of row 20. These percentages are used to compare andconvert between fractions, decimals, and percentages.

[0167] For example: {fraction (1/10)}² to 10²/10², {fraction (1/100)} to{fraction (100/100)}, 0.01 to 1, and 1% to 100%

[0168] (10) Relation among arrays:

[0169] (Referring to FIG. 2, FIG. 2A, and FIG. 2B).

[0170] Relation between arrays are based on power of ten ,10^(a), where“a” is an integer with range from minus infinitive to plus infinitive({−∞, +∞}). Power of ten “10^(a”) is used in a plane with grid atbackground. One of horizontal line of the grid is used as horizontalaxis. Power of ten 10⁰ is placed at an intersection of the horizontalaxis and a vertical line of the grid. In the plane, increasingdirections of powers of ten are to the left and up or to the top. Fromlocation of 10⁰ on the horizontal axis, where “a” is equal to zero,power of ten “10^(a)” increases to infinite (∞) to the left and at thesame time, decreases to minus infinite (−∞) to opposite direction, tothe right. Each power of ten is located at an intersection of horizontalaxis with each vertical line of the grid, consecutively, as shown belowand in FIG. 2B:

[0171] . . . 10⁵ 10⁴ 10³ 10² 10¹ 10⁰ 10⁻¹ 10⁻² 10⁻³ 10⁻⁴ 10⁻⁵ . . .

[0172] Then all the powers of ten on the horizontal axis both increaseup to the top, and decrease down to bottom at same time such that powersof ten are located at intersections of horizontal lines with verticallines of the grid. Thus, an array of powers of ten is formed with powersof ten consecutively ranging from minus infinitive to plus infinitive indirection from right to left on horizontal rows, and from bottom to topon vertical rows. Infinite squares are formed and each square is definedby four powers of ten, “10^(a)”, “10^(a+1)”, “10^(a+2)”, and 10^(a+1)”,at four corners, bottom right, top right, top left, and bottom leftcorner, respectively, as shown in FIG. 2B with four powers of tenhigh-lighted. Increasing directions, up and to the left, make number(decimal form for power of ten ) at bottom right corner in each squareone-hundredth of the number at top left corner, and the number at topright corner equal to the number at bottom left corner. At any power often “10^(a)”, next number to the left or up is “10^(a+)”, and nextnumber to the right or down is “10^(a−1)”. The repeated addition ofnumber “10⁰” up and/or to the left, as well as the repeated addition ofother numbers are used to form big numbers in array 24 (FIG. 2). Smallnumbers and others are added to array 24 for other purposes of thepreferred embodiment as explained in ‘Operation’ from (1) to (9).

[0173] ** Array 24: (Referring to ‘Operation’, (1) to (6))

[0174] ** Arrays 12,16, 32, and 36:

[0175] In these arrays, upper or lower number in each square space isone-tenth of the upper or lower number in next square space straight upor to the left in same array or in other array, or ten times of theupper or lower number in next square space straight down or to the rightin same array or in other array, respectively.

[0176] For numbers in square spaces of column 44 and row 20, big numberin each square space is used in according with upper or lower number innext square space up, down, to the left or to the right in other arrays.

[0177] ** Arrays 14 and 34:

[0178] Each big number in each square space of column 44, portions inarray 14 and 34, is used in repeated addition by itself to the left to anumber, ten times of it and in same row, in square space of firstcolumns from left of these arrays. So, upper or lower number in eachsquare space of arrays 14 and 34 is one-tenth of the upper or lowernumber in next square space straight up in same array, and in nextsquare space straight up or to the left in other arrays, respectively.Upper or lower number in each square space is ten times of the upper orlower number in next square space straight down in same array, and innext square space straight down or to the right in other arrays,respectively.

[0179] ** Arrays 26 and 22:

[0180] Each big number in each square space of row 20, portions in array22 and 26, is used in repeated addition by itself to the top to anumber, ten times of it and in same column, in square space of firstrows from top of these arrays. So, upper or lower number in each squarespace of arrays 22 and 26 is one-tenth of the upper or lower number innext square space straight to the left in same array, and in next squarespace straight to the left or up in other arrays, respectively. Upper orlower number in each square space is ten times of the upper or lowernumber in next square space straight to the right in same array, and innext square space straight to the right or down in other arrays,respectively.

[0181] For numbers in square spaces of array 24, big number in eachsquare space of outer rows and columns is used in according with upperor lower number in next square space up, down, to the left or to theright, in other arrays.

[0182] Each square space in each array has a symmetrical square spacethrough diagonal line from bottom right vertex of array 12 to top leftvertex of array 36, excluding square spaces situated on that line.

[0183] Operation for The First and Second Alternative Embodiments

[0184] (FIG. 3, FIG. 4, FIG. 2A, and FIG. 2B)

[0185] Operation for the first and second alternative embodiments is insame manner with operation for the preferred embodiment.

[0186] Column 44 and row 20 of the first alternative embodiment (FIG. 3)are added to mainly make room for small numbers at corners, dashes, anddots in square spaces of column 44 and row 20 of the preferredembodiment to move to, so the operation does not change. The phrase:“Excluding square spaces with big numbers in column 44 and row 20 . . .” is used where applicable.

[0187] With column and row of zero's (0's), the second alternativeembodiment (FIG. 4) has some minor differences in operation. Forexample: In subtraction operation for numbers in the same row,difference of the big number as minuend and small number on the right assubtrahend in each square space of third column from right is zero andshown (compared with “not shown” in ‘Operation’ for the preferredembodiment). Number zero (0) used in finding whole numbers is shown,compared with “not shown” in section ‘7 Whole numbers’ of ‘Operation’for the preferred embodiment. The phrase: “Excluding square spaces withbig numbers in column 44 and row 20, and zeros in . . .” is used whereapplicable.

[0188] (f) Conclusion and Ramification:

[0189] (FIG. 2, FIG. 3, FIG. 4, FIG. 2A, FIG. 2B, and FIG. 2C)

[0190] In consideration of the drawings and description above, it isapparent that the basic mathematics teaching and learning aid of thepresent invention can be used as an aid to teach and learn basicmathematics simply, clearly, easily, and conveniently. In anunprecedented way, it concretely uses what is known to teach what isunknown. It mathematically provides simple steps to show how one thingis made from another one in basic mathematics, and basically how theyare related to one another. By that way, it is much easier for theusers, especially the young ones, to learn basic mathematics.

[0191] In order to help illustrate the preferred embodiment, alternativeembodiments, and some aspects in constructing the present invention, thedescription above relates the present invention to many specificitiesThese should not be considered as limits to the scope of the presentinvention. The size, color, arrangement, and uses of the preferredembodiment presented in the description can be adjusted, changed ormodified in such a way that is still within the scope of presentinvention. For examples:

[0192] All arrays 12,14,16,22,26,32,34, and 36 can be moved closer to orfather from array 24. Other symbols for multiplication and division canbe used. The symbols of addition, subtraction, multiplication, anddivision, as well as the arrows can be placed anywhere in open spaces40, 42, and 18, in such a way that facilitates understanding of theusers.

[0193] Square spaces in each array can be separated from one another byan equal substantial distance. Small numbers in each square space ofarray 24 can be relocated at other positions in same square space oroutside in open space among square spaces.

[0194] The square spaces in arrays used to convey information can be inany shape, such as polygon, circular, oval, etc.

[0195] Increasing direction of number “10^(a)” in a plane or inthree-dimensional space can be of any direction.

[0196] Two or four squares, each defined by four powers often, “10^(a)”,“10^(a+1)”, “10^(a+2)”, and “10^(a+1)” at four corners with expandingnumbers like array 24, next to each other can be presented in anembodiment.

[0197] The preferred embodiment can be modified to apply inthree-dimensional space as illustrated in FIG. 2C.

[0198] Multiples of each number in open spaces 42 and 18, exceptpercentages, can be accordingly placed next to that number as long asthose open spaces have enough room.

[0199] In the second alternative embodiment (FIG. 4), small numbers canbe placed in each square space with big number zero (0) of array 24 andof other arrays with compatible using.

[0200] The preferred embodiment can be built with enough thickness sothat lights can be put under transparent surfaces of square spaces inarrays, and controlled by the users corresponding to functions of thepresent invention.

[0201] Numbers, letters, dashes, dots, arrows, percentages, symbols andsign in the preferred embodiment can be raised or replaced with raisedperiods (dots) in Braille for the handicapped to use the presentinvention.

[0202] The preferred embodiment can be use to present numbers (1, 2, 3,. . . ) in base-ten number system to kindergarten students.

[0203] With some modifications, the preferred embodiment can be used aseducational game within the scope of the present invention.

[0204] Instead of in suitable size for personal use, the preferredembodiment can be made bigger with different suitable material to usewith group of users such as in classroom, etc.

[0205] Numbers in each square space (and with numbers in other squarespaces) are ready for “homework” or practice with mathematical problemsand solutions.

[0206] The preferred embodiment can be generated by a computer programso that it can be used on screen of a computer. Then, a click ofcomputer mouse on a square space of the preferred embodiment will showhow numbers in that square space and with numbers in other square spacesperform mathematical operations, and how they are related to oneanother. Explanations and answers for questions are also made available.

[0207] The preferred embodiment can also be used to replace calculatorin daily simple calculations.

[0208] Accordingly, the scope of the invention should be determined, notby the specificities in the description and examples given above, but bythe appended claims and their legal equivalents.

I claim:
 1. A basic mathematics teaching and learning aid comprising aportion of squares each defined by four powers of ten, 10^(a), 10^(a+1),10^(a+2), and 10^(a+1) where power a being an integer in the range fromminus infinitive to plus infinitive with consecutively increasingdirections being to the left and to the top, at four corners, bottomright, top right, top left and bottom left corner of the squarerespectively; each said square placed in an array of spaces with saidfour powers of ten situating in four spaces at four corners of thearray, bottom right, top right, top left, and bottom left cornerrespectively;
 2. The basic mathematics teaching and learning aid ofclaim 1 wherein said array with four powers of ten, 10⁰, 10¹, 10², and10¹ comprises: (a) numbers of different sizes in each space of the arraywith number nearly at center substantial bigger than numbers on theright and on bottom; (b) small number on the right of big number in eachof spaces of same row forming line of same small numbers, and being samewith big number in space of outer right column in same row; (c) smallnumber on bottom of big number in each of spaces of same column formingline of same small numbers, and being same with big number in space ofbottom row in same column; (d) each space of outer right column andbottom row of the array, except common space of the column and row,having a dot at bottom right corner, a substantial small number atbottom left corner which has value equal to value of the big number insame space minus one, a substantial small number at top left cornerwhich is same with the big number, a substantial small number 1 at topright corner, and a dash between the small numbers at top left and topright corners; the common space of the row and column having a dot atbottom right corner, a substantial small number 0 at bottom left corner,and two substantial small number 1's at top left and top right corners;(e) the big numbers and small numbers in spaces of the array presentingin decimal form;
 3. The basic mathematics teaching and learning aid ofclaim 1 wherein said array with four powers of ten, 10⁻², 10⁻¹, 10⁰, and10⁻¹ comprises: (a) spaces of outer right column each having numbers ofdifferent sizes with number nearly at center substantial bigger thannumbers at corners; a dot at bottom left corner, a small number 1 at topleft corner, a small number at top right corner which is same with thebig number, and a dash between the small numbers at top left and topright corners; (b) number in each space of the array, except spaces ofthe outer right column, presenting in decimal form and in exponentialform;
 4. The basic mathematics teaching and learning aid of claim 1wherein said array with four powers of ten, 10⁻², 10⁻¹, 10⁰, and 10⁻¹comprises: (a) spaces of bottom row each having numbers of differentsizes with number nearly at center substantial bigger than numbers atcorners; a dot at bottom left corner, a small number 1 at top leftcorner, a small number at top right corner which is same with the bignumber, and a dash between the small numbers at top left and top rightcorners; (b) number in each space of the array, except spaces of thebottom row, presenting in decimal form and in exponential form;
 5. Thebasic mathematics teaching and learning aid of claim 1 wherein saidarray with four powers of ten, 10², 10³, 10⁴, and 10³ comprises: (a)spaces of bottom row each having numbers of different sizes with numbernearly at center substantial bigger than numbers at corners; a dot atbottom right corner, a small number at bottom left corner which hasvalue equal to value of the big number in same space minus one and is indecimal form, a small number 1 at top right corner, a small number attop left corner which is same with the big number, and a dash betweenthe small numbers at top left and top right corners; (b) number in eachspace of the array, except spaces of the bottom row, presenting indecimal form and in exponential form;
 6. The basic mathematics teachingand learning aid of claim 1 wherein said array with four powers of ten,10²,10³,10⁴, and 10³ comprises: (a) spaces of outer right column eachhaving numbers of different sizes with number nearly at centersubstantial bigger than numbers at corners; a dot at bottom rightcorner, a small number at bottom left corner which has value equal tovalue of the big number in same space minus one and is in decimal form,a small number 1 at top right corner, a small number at top left cornerwhich is same with the big number, and a dash between the small numbersat top left and top right corners; (b) number in each space of thearray, except spaces of the outer right column, presenting in decimalform and in exponential form;
 7. The basic mathematics teaching andlearning aid of claim 1 wherein said arrays each having four powers often each, 10⁻⁴, 10⁻³, 10⁻², 10⁻³, and 10⁰, 10¹, 10², 10¹, and 10⁰, 10¹,10², 10¹, and 10⁴, 10⁵, 10⁶, 10⁵ have number in each space of the arrayspresenting in decimal form and in exponential form;
 8. The basicmathematics teaching and learning aid of claim 1 wherein said arrays areseparated from one another by a substantial distance referred as openspace;
 9. The basic mathematics teaching and learning aid of claim 8wherein said open space comprises: (a) mathematical symbols of addition,subtraction, multiplication, and division; (b) sign formed by two shortvertical and horizontal lines and arrows; (c) percentages of 100 and 1;(d) numbers, in decimal form, presenting quotients of 1 as dividend witheach big number in spaces of outer right columns and bottom rows of saidbottom, top, right, center, and left arrays as divisor, except bignumber 1 in common space; (e) equations of a number of n with zero,n+0=n, n−0=n , 0×n=0 , and 0/n=0 and condition n≠0 for the latestequation;
 10. A basic mathematics teaching and learning aid comprisingnine arrays of spaces referred as bottom right, bottom, bottom left,right, center, left, top right, top, and top left arrays, each placedwith four powers of ten, 10^(a), 10^(a+1), 10^(a+2), and 10^(a+1) wherepower a being an integer in the range from −4 to 6 with consecutivelyincreasing directions being to the left and to the top, in four spacesat four corners, bottom right, top right, top left, and bottom leftcorner of the array, respectively;
 11. The basic mathematics teachingand learning aid of claim 10 wherein said center array comprises: (a)numbers of different sizes in each space of the array with number nearlyat center substantial bigger than numbers on the right and on bottom;(b) small number on the right of big number in each of spaces of samerow forming line of same small numbers, and being same with big numberin space of outer right column in same row; (c) small number on bottomof big number in each of spaces of same column forming line of samesmall numbers, and being same with big number in space of bottom row insame column; (d) each space of outer right column and bottom row of thearray, except common space of the column and row, having a dot at bottomright corner, a substantial small number at bottom left corner which hasvalue equal to value of the big number in same space minus one, asubstantial small number at top left corner which is same with the bignumber, a substantial small number 1 at top right corner, and a dashbetween the small numbers at top left and top right corners; the commonspace of the row and column having a dot at bottom right corner, asubstantial small number 0 at bottom left corner, and two substantialsmall number 1's at top left and top right corners; (e) the big numbersand small numbers in spaces of the array presenting in decimal form; 12.The basic mathematics teaching and learning aid of claim 10 wherein saidbottom array comprises: (a) spaces of outer right column each havingnumbers of different sizes with number nearly at center substantialbigger than numbers at corners; a dot at bottom left corner, a smallnumber 1 at top left corner, a small number at top right corner which issame with the big number, and a dash between the small numbers at topleft and top right corners; (b) number in each space of the array,except spaces of the outer right column, presenting in decimal form andin exponential form;
 13. The basic mathematics teaching and learning aidof claim 10 wherein said right array comprises: (a) spaces of bottom roweach having numbers of different sizes with number nearly at centersubstantial bigger than numbers at corners; a dot at bottom left corner,a small number 1 at top left corner, a small number at top right cornerwhich is same with the big number, and a dash between the small numbersat top left and top right corners; (b) number in each space of thearray, except spaces of the bottom row, presenting in decimal form andin exponential form;
 14. The basic mathematics teaching and learning aidof claim 10 wherein said left array comprises: (a) spaces of bottom roweach having numbers of different sizes with number nearly at centersubstantial bigger than numbers at corners; a dot at bottom rightcorner, a small number at bottom left corner which has value equal tovalue of the big number in same space minus one and is in decimal form,a small number 1 at top right corner, a small number at top left cornerwhich is same with the big number, and a dash between the small numbersat top left and top right corners; (b) number in each space of thearray, except spaces of the bottom row, presenting in decimal form andin exponential form;
 15. The basic mathematics teaching and learning aidof claim 10 wherein said top array comprises: (a) spaces of outer rightcolumn each having numbers of different sizes with number nearly atcenter substantial bigger than numbers at corners; a dot at bottom rightcorner, a small number at bottom left corner which has value equal tovalue of the big number in same space minus one and is in decimal form,a small number 1 at top right corner, a small number at top left cornerwhich is same with the big number, and a dash between the small numbersat top left and top right corners; (b) number in each space of thearray, except spaces of the outer right column, presenting in decimalform and in exponential form;
 16. The basic mathematics teaching andlearning aid of claim 10 wherein said bottom right, bottom left, topright, and top left arrays have number in each space of the arrayspresenting in decimal form and in exponential form;
 17. The basicmathematics teaching and learning aid of claim 10 wherein said arraysare separated from one another by a substantial distance referred asopen space;
 18. The basic mathematics teaching and learning aid of claim17 wherein open space among said arrays comprises: (a) mathematicalsymbols of addition, subtraction, multiplication, and division; (b) signformed by two short vertical and horizontal lines and arrows; (c)percentages of 100 and 1; (d) numbers, in decimal form, presentingquotients of 1 as dividend with each big number in spaces of outer rightcolumns and bottom rows of said bottom, top, right, center, and leftarrays as divisor, except big number 1 in common space; (e) equations ofa number of n with zero, n+0=n, n−0=n, 0×n=0, and 0/n=0 and conditionn≠0 for the latest equation;
 19. A basic mathematics teaching andlearning aid comprising: a plurality of arrays of spaces, each arrayplaced with four powers of ten, 10^(a), 10^(a+1), 10^(a+2), and 10^(a+1)where power a being an integer in the range from minus infinitive toplus infinitive with consecutively increasing directions being to theleft and to the top, in four spaces at four corners, bottom right, topright, top left, and bottom left corner of the array, respectively,numbers in decimal form and of different sizes in some of said spaces;numbers in decimal form and of different sizes and dot and dash in someof other said spaces; numbers in decimal form and in exponential form insome of still other said spaces; arrows, sign, and mathematical symbolsof addition, subtraction, multiplication, and division situating in openspace among said arrays; numbers, in decimal form, presenting quotientsof 1 as dividend with each of some of big numbers in some of said spacesas divisor situating in said open space next to each said space witheach of the big numbers, accordingly; equations of a number of n withzero in n+0=n , n−0=n , 0×n=0 , and 0/n=0 condition n≠0 for the latestequation situating in said open space; percentages of 100 and 1 withlittle horizontal line in between as well as percentages of 100 and 1with a little vertical line in between situating in said open space;